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Learning Math Home
Patterns, Functions, and Algebra
 
Session 8 Part A Part B Part C Homework
 
Glossary
Algebra Site Map
Session 8 Materials:
Notes
Solutions
 

A B C 
Homework

Video

Solutions for Session 8, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6| B7 | B8 | B9 |
B10 | B11 | B12 | B13


Problem B1

Here is the completed table:

 

Number of teachers

 

Amount each
will receive

1

 

800,000

2

 

400,000

3

 

266,666.66

4

 

200,000

10

 

80,000

15

 

53,333.33

20

 

40,000

100

 

8,000

 

The last two entries will vary.

<< back to Problem B1


 

Problem B2

One rule is that the product of the input and the output is always 800,000. If M = the money each teacher receives, and T = the number of teachers, then
(M) x (T) = 800,000. This is because the total prize of 800,000 is split evenly among the teachers. This also leads to a second rule: M = 800,000 / T, because the money received by each teacher is 800,000 divided by how many teachers split the prize. Additionally, T = 800,000 / M.

<< back to Problem B2


 

Problem B3

The amount of money decreases as the number of teachers increases, but it is not an exponential decay because the ratio between consecutive outputs is not constant. Looking at differences between outputs guarantees that the graph is neither linear nor quadratic. The graph is not cyclic, because there is no point where the outputs begin to repeat.

<< back to Problem B3


 

Problem B4

Here is the completed table:

 

Length

 

Width

 

Area = length * width

50

 

40

 

2000

25

 

80

 

2000

100

 

20

 

2000

1,000

 

2

 

2000

40

 

50

 

2000

80

 

25

 

2000

2000

 

1

 

2000

0.5

 

4000

 

2000

 

The last four entries will vary.

<< back to Problem B4

<< back to Problem B4 low-tech version


 

Problem B5

One equation is x * y = 2,000. Note that Area = length * width is always constant at 2,000. Other possible equations are y = 2,000 / x and x = 2,000 / y.

<< back to Problem B5

<< back to Problem B5 low-tech version


 

Problem B6

<< back to Problem B6

<< back to Problem B6 low-tech version


 

Problem B7

Overall, if x is multiplied by a number, y is divided by the same number. If x is divided by a number, y is multiplied by the same number.

a. 

Here is the completed table. All y-values are rounded to one decimal place.

 

x

 

y

 

Decrease in y

20

 

100

 

--

30

 

66.7

 

33.3

40

 

50

 

16.7

50

 

40

 

10

60

 

33.3

 

6.7

70

 

28.6

 

4.7

80

 

25

 

3.6

90

 

22.2

 

2.8

100

 

20

 

2.2

 

b. 

As x increases by 10, y continuously decreases, but the rate of decrease lessens as x grows.

c. 

If x doubles, y is cut in half. If x triples, y is divided by three.

d. 

If x is very small, y must be very large, since the product of x and y is always the same. By the same argument, if x is very large, then y must be very small.

<< back to Problem B7

<< back to Problem B7 low-tech version


 

Problem B8

Let's try it:

 

x

 

y

-10

 

-0.3

-6

 

-0.5

-3

 

-1

-2

 

-1.5

-0.5

 

-6

0.5

 

6

2

 

1.5

3

 

1

6

 

0.5

10

 

0.3

 

<< back to Problem B8


 

Problem B9

If x = 0, y is undefined, which means that no value of y will make zero times y equal to 3. Using a calculator, 3 / 0 will fail to return a number; the calculator will give an error message.

<< back to Problem B9


 

Problem B10

The graph will not cross the y-axis, because if it did, it would mean that some y-value would be assigned for x = 0 for the graph, and, as explained in the solution to Problem B9, there is no such value. Another way to look at it is to examine the behavior of the graph near x = 0. If x is a little more than zero, y is a very large, positive number, but if x is a little less than zero, y is a very large, negative number. These portions of the graph do not meet.

<< back to Problem B10


 

Problem B11

The 3 represents the area of a rectangle drawn with (0, 0) as one corner and any point on the graph as the opposite corner. Compare this to the Interactive Activity, where there were rectangles of area equaling 2,000 feet -- length times width always equaled 2,000. Here, x is the length, and y is the width.

<< back to Problem B11


 

Problem B12

In a direct variation function, an increase in x creates a proportional increase in y; if x is multiplied by 5, y is also multiplied by 5. But in inverse variation, the opposite is true: If x is multiplied by 5, y is divided by 5. The word "inverse" refers to the inverse operations of multiplication and division.

<< back to Problem B12


 

Problem B13

In this particular equation, x is the reciprocal of y, because x and y multiply together to make 1. The reciprocal is used to solve equations like 5n = 16 -- multiplying both sides of the equation by the reciprocal of 5 produces two numbers (5 and 1/5) which, when multiplied, make 1. So, multiplying by 1/5 will remove the 5 from the left side, leaving variable n by itself. This is the key to solving equations by backtracking.

<< back to Problem B13


 

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